determining the refractive index dispersion and thickness ... · method to determine the refractive...

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Determining the refractive index dispersionand thickness of hot-pressed chalcogenide thin filmsfrom an improved Swanepoel method

Y. Fang1• D. Jayasuriya1

• D. Furniss1• Z. Q. Tang1

•

Ł. Sojka1,2• C. Markos1,3

• S. Sujecki1,2• A. B. Seddon1

•

T. M. Benson1

Received: 7 March 2017 / Accepted: 16 May 2017 / Published online: 3 June 2017� The Author(s) 2017. This article is an open access publication

Abstract The well-known method presented by Swanepoel can be used to determine the

refractive index dispersion of thin films in the near-infrared region from wavelength values

at maxima and minima, only, of the transmission interference fringes. In order to extend

this method into the mid-infrared spectral region (our measurements are over the wave-

length range from 2 to 25 lm), the method is improved by using a two-term Sellmeier

model instead of the Cauchy model as the dispersive equation. Chalcogenide thin films of

nominal batch composition As40Se60 (at.%) and Ge16As24Se15.5Te44.5 (at.%) are prepared

by a hot-pressing technique. The refractive index dispersion of the chalcogenide thin films

is determined by the improved method with a standard deviation of less than 0.0027. The

accuracy of the method is shown to be better than 0.4% at a wavelength of 3.1 lm by

comparison with a benchmark refractive index value obtained from prism measurements

on Ge16As24Se15.5Te44.5 material taken from the same batch.

Keywords Chalcogenide glasses � Refractive index � Dispersion

1 Introduction

A chalcogenide glass is a glass containing one or more Group 16 elements of the Periodic

Table, sulfur (S), selenium (Se), or tellurium (Te), usually formulated with additions of

germanium, arsenic or antimony to increase glass stability and robustness (Seddon 1995).

& T. M. [email protected]

1 Mid-Infrared Photonics Group, George Green Institute for Electromagnetics Research, Faculty ofEngineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK

2 Institute of Telecommunication, Teleinformatics and Acoustics, Wroclaw University ofTechnology, Wybrzeze Wyspianskiego 27, 50-370 Wrocław, Poland

3 DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark,Ørsteds Plads 343, 2800 Kongens Lyngby, Denmark

123

Opt Quant Electron (2017) 49:237DOI 10.1007/s11082-017-1057-9

http://orcid.org/0000-0002-1632-1816

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Since chalcogenide glasses provide a wide range of glass compositions, different physical

properties can be achieved, including refractive index. The properties of mid-infrared

(MIR) transparency, high refractive index, low phonon energy, high optical non-linearity

(Zakery and Elliott 2003), and an ability to doped them with rare-earth element ions

(Sanghera and Aggarwal 1999; Sanghera et al. 2009; Tang et al. 2015; Falconi et al. 2016),

make chalcogenide glasses attractive for use in planar photonic integrated circuits (Seddon

et al. 2006, 2014; Abdel-Moneim et al. 2015), and narrow- and broad-band fibre-based

laser sources (Petersen et al. 2014) and amplifiers (Hu et al. 2015; Falconi et al. 2017) for

the MIR. Although much research effort has been paid to the development and charac-

terisation of chalcogenide glasses for photonics, relatively little refractive index dispersion

data are presently available at MIR wavelengths, see for example: Orava et al. (2009), Qiao

et al. (2011), Dantanarayana et al. (2014), Gleason et al. (2016), Wang et al. (2017) and

references therein.

Refractive index, and very often its wavelength dispersion, is one of the key parameters

that influences the design of optical components. Techniques commonly used for refractive

index measurement include the modelling of transmission or reflection spectroscopic data

(Swanepoel 1985; Marquez et al. 1998), spectroscopic ellipsometry (Orava et al. 2009;

Wang et al. 2017), prism coupling (Carlie et al. 2011), grating coupling (Laniel et al.

2003), and the measurement of the minimum deviation angle of light passing through a

prism (Fang et al. 2016). Several of these techniques require intensive sample preparation,

are time consuming, and are costly to conduct. More importantly, it has until recently been

difficult to apply some of these techniques to the MIR spectral region because of the

paucity of suitable beam sources (Carlie et al. 2011). Carlie et al. (2011) modified a

commercial system prism coupling system (Metricon model 2010) using additional laser

sources, detectors, and a GaP prism in order to enable the measurement of refractive index

dispersion at spot wavelengths over the 1.5–10.6 lm range and used it to determine the

refractive index of As2Se3 bulk glass and thin films across this wavelength region to an

accuracy of ±0.001. Due to the high accuracy of the method the authors were able to

compare the refractive indices of as-deposited and annealed As2Se3 films with those of the

parent bulk glasses. They showed that an as-deposited thin film had a lower refractive

index than that of the bulk; subsequent annealing of the films increased the refractive index

back up to the level of the bulk glass. This illustrates the importance of the thermal history

of the glass on its physical structure, and hence on its physical properties such as density

and refractive index. In Dantanarayana et al. (2014) we reported the continuous linear

refractive index dispersion at ambient temperature of two bulk chalcogenide glasses, AsSe

and GeAsSe, over the 0.4–33 lm wavelength range obtained by means of spectroscopic

ellipsometry. A two-term Sellmeier equation, with one resonant visible-region absorption

and one resonant far-infrared absorption, was determined as sufficient and appropriate for

fitting the refractive index dispersion over the wavelength range for which the extinction

coefficient was less than 0.0005. The fitting accuracy was better than 0.1%. Nevertheless,

the accuracy in measuring the refractive index by means of spectroscopic ellipsometry is

inherently limited by surface effects (Poelman and Smet 2003) as a thin contamination

layer, oxide layer, or small defects on the surface can all potentially yield different optical

constants for the glass.

The method of Swanepoel (1985) was presented to determine the refractive index and

thickness of thin films in the transparent region from wavelength values at maxima and

minima, only, of the transmission interference fringes. This entails measuring two trans-

mission spectra, one at normal incidence and another at oblique incidence. The Swanepoel

method was used by Corrales et al. (1995) to measure the refractive index of an As2Se3 thin

237 Page 2 of 19 Y. Fang et al.

123

film in the 600–2000 nm wavelength region and the method yielded values for refractive

index and average thin film thickness to an accuracy of better than 3%. In this paper, an

improved Swanepoel method, based on Swanepoel (1985), but modified by the introduc-

tion of a Sellmeier dispersion equation in order improve the applicability of the method to

chalcogenide glasses at MIR wavelengths, is introduced. It is used to determine the

refractive index dispersion of thin As40Se60 (at.%, nominal batch composition) and

Ge16As24Se15.5Te44.5 (at.%, nominal batch composition) hot-pressed glass films in the MIR

spectral region to an accuracy of better than 0.4%. The actual glass compositions measured

using scanning electron microscopy energy dispersive X-ray diffraction (SEM-EDX) were

found to match the nominal batch compositions to within the ±0.7 at.% accuracy and

precision of the measurements. The present improvements apply to the transparency region

of chalcogenide glasses and are distinct from other improvements to the Swanepoel

method recently suggested by Jin et al. (2017) who described on a tangency point method

(TPM) to resolve problems observed when using the Swanepoel method in the strong

absorption region.

The remainder of the paper is organised in the following way: in Sect. 2, the preparation

of thin film samples of Ge16As24Se15.5Te44.5 and As40Se60 is introduced, which embraces

glass-melting, fibre-drawing, propylamine-etching and hot-pressing. This section also

presents experimental measurements of the transmission through the GeAsSeTe and AsSe

thin films using Fourier transform infrared spectroscopy (FTIR). Section 3 discusses the

appropriate refractive index model to describe the refractive index dispersion of chalco-

genide glasses in the MIR spectral region and the basic theory of the improved Swanepoel

method to determine the refractive index and thickness of the thin films. Section 4 presents

the refractive index and thickness of the thin films obtained from the improved Swanepoel

method. The improvements in determining the refractive index dispersion from the

improved Swanepoel method are highlighted by comparison with the original approach.

The results from the improved method are also compared to the results from ellipsometry,

the minimum deviation method and commercial data, in this section. The important

conclusions drawn from the study are highlighted in Sect. 5.

2 Sample preparation and FTIR measurements

2.1 Glass melting

In preparing the glass Ge16As24Se15.5Te44.5 and As40Se60 compositions studied here, the As

(7 N purity, Furakawa Denshi) and Se (5 N purity, Materion) elemental precursors were

heat-treated under vacuum (10-3 Pa) at 310 and 260 �C, respectively, in order to purify

them by driving off volatile oxides. Ge (5 N purity, Materion) and Te (5 N purity,

Materion) were untreated. The elements were subsequently batched under a nitrogen

atmosphere in a MBraun glove-box (B0.1 ppm H2O and B0.1 ppm O2) into a vitreous

silica ampoule (Multi-Lab Ltd, UK). Before batching the elements, the vitreous silica

ampoule was etched using 40% v/v hydrofluoric acid (HF) and rinsed with deionised water;

it was next air and then vacuum baked (under vacuum *10-3 Pa) at 1000 �C for 6 h in a

vertical furnace (TF105/4.5/IZF, Instron) to drive off physisorbed and chemisorbed water

from the inside surface of the silica glass ampoule. After batching, the silica ampoule was

sealed (under *10-3 Pa vacuum) and placed in a rocking furnace to melt and homogenise

the glass, at *750 �C/12 h for the Ge16As24Se15.5Te44.5 composition and at *800 �C/12 h

Determining the refractive index dispersion and thickness… Page 3 of 19 237

123

for the As40Se60 composition. The melts were quenched and annealed for 1 h at their

respective glass transition temperature (Tg) before slowly cooling them down to room

temperature. This produced glass rods which were *10 mm in diameter and *70 mm in

length.

2.2 Thin film sample preparation

A hot-pressing technique was developed to prepare thin films of chalcogenide glass; this

involved the hot-pressing of circa 20 mm lengths of fibre, with a nominal diameter of

250 lm, using the hot-press previously described in Fang et al. (2016) (ICTON): Seddon

et al. (2006), Abdel-Moneim et al. (2015). The as-annealed preforms (*10 mm diameter

and *70 mm length) of Ge16As24Se15.5Te44.5 and As40Se60, prepared as in Sect. 2.1, were

drawn into optical fibres using a customised Heathway draw-tower in a 10,000-Class

clean-room. The fibres were drawn under a N2 flow (BOC white-spot) at a viscosity of

around 104.5 Pa s. The final fibre diameter was 250 ± 10 lm. The outer surface of the

fibres was shiny and no defects could be seen by the naked eye. Five Ge16As24Se15.5Te44.5

fibres, each of about 250 lm diameter and 20 mm length, were cleaved from the Ge16-

As24Se15.5Te44.5 fibre. The five 20 mm lengths of chalcogenide fibre were each immersed

in propylamine (Sigma-Aldrich 99 mol.%) for 30 min at room temperature under atmo-

spheric pressure. Subsequently the etched fibres were taken out of the propylamine and

were immersed in acetone (99.8%, Sigma Aldrich) for 10 min and subsequently each fibre

was individually cleaned carefully using isopropanol (IPA, HPLC grade, Fisher Chemical)

with optical lens tissue. A silica glass ampoule (Multi-Lab Ltd, UK) was used as the

container for the propylamine-etching; this was rinsed with distilled water and dried under

vacuum (*10-3 Pa). The five propylamine-etched 20 mm long fibres were then placed on

a tungsten carbide plate (800 mm diameter, of flatness 0.08 lm and of surface finish

0.009 lm) surrounded by a 20 lm thick stainless steel shim (Supplier: Hollinbrow Ltd,

UK) and then covered by a second tungsten carbide plate. The fibre samples were hot-

pressed under vacuum (10-4 Pa) at Tg ? 40 �C (viscosity *108 Pa s) with a maximum

pressure of 700 N between the two plates. The samples were then annealed at the onset-Tg

(175 �C) and allowed to cool with the discs. In order not to damage the samples while

opening the press, solvents (acetone and if necessary IPA) were squirted into the gap

between the two tungsten carbide plates. As40Se60 thin films, this time with a 25 lm target

(shim) thickness, were also prepared following the same method.

2.3 Optical transmission measurements on the thin-film chalcogenide glasssamples using FTIR

A GlobarTM source, KBr beam splitter and DLaTGSD301 detector were set up in the FTIR

spectrometer (Bruker IFS 66v/s) to measure the interference transmission spectra of the

thin films in the wavelength range from 1 to 25 lm; 25 lm was the maximum wavelength

possible with this set-up in the Bruker FTIR. Because of the low efficiency of the KBr

beam splitter and the DLaTGSD301 detector at wavelengths below 2 lm, clear extrema of

interference fringes were only obtained over the wavelength range from 2 to 25 lm.

The Ge16As24Se15.5Te44.5 or As40Se60 thin film prepared using the hot-pressing tech-

nique was mounted on a home-made rotation stage, which was accurate up to 0.1�,installed in the sample compartment of the FTIR spectrometer. In the FTIR spectrometer a

He–Ne beam, which was used for alignment, indicated the position of the GlobarTM.


123

Careful alignment was undertaken to ensure that the axis of rotation went through the

surface of the thin film at the focal point of the He–Ne beam and that no lateral dis-

placement of the light took place during rotation. The location of normal incidence was

determined by ensuring that equivalent transmission spectra were obtained after rotation

through equal angles in a clockwise and anticlockwise sense.

In FTIR thin film transmission measurements, air absorption in the optical path in the

mid-infrared region distorted the observed interference fringes, as indicated in Fig. 1. The

main absorption bands observed were: CO2 absorption at around 4.2 and 15 lm and H–O–

H absorption at around 2.7 and 6.2 lm (Gibson and MacGregor 2013). A background

spectrum was collected immediately before each thin film transmission measurement,

without the sample present, with care to ensure that the measurement conditions remained

the same; this considerably reduced the effect of air absorption.

Figure 2a, b shows the FTIR interference transmission spectra at normal incidence and

at an incident angle of 30� of the Ge16As24Se15.5Te44.5 and As40Se60 thin films, respec-

tively. In Fig. 2, the solid curves are the interference fringes obtained from the case of

normal incidence and the red curves are from the case of oblique incidence. For the

transmission spectra of As40Se60 in Fig. 2b, an obvious material absorption occurred at

*21 lm, which was identified in Maklad et al. (1974) as the second overtone of the

fundamental stretching band due to AsSe3 pyramid units in a As2Se3 glass structural matrix

itself. This will be discussed further in Sect. 4 with regards to the Sellmeier model adopted

in this paper.

3 The theory and analysis of the improved Swanepoel method

3.1 The dispersive refractive index model

Referring to Fig. 2, the method of Swanepoel (1985) was developed to determine the

dispersive refractive index and thickness of thin films in the region where k2 � n2 from

wavelength values at maxima and minima, only, of the transmission interference fringe.

This entailed obtaining two spectra, one at normal incidence and another at oblique

incidence. In Swanepoel (1985) the material refractive index was fitted to the dispersive

equation given in Eq. (1):

0.4

0.5

0.6

0.7

0.8

0.9

1

2 3 4 5 6 7 8

Tran

smitt

ance

Wavelength /µm

CO2 absorption band at circa 4.2 μm

Fig. 1 The transmission spectrum of a Ge16As24Se15.5Te44.5 thin film of nominal thickness 20 lm withoutcareful background collection


123

n2 ¼ Aþ B

kxð1Þ

where k is wavelength and A, B and x are fitting coefficients. The dispersive equation in

Eq. (1) is similar to the Cauchy equation, but in order to have a better fit to the refractive

index data, the Cauchy model was altered to have an extra fitting parameter x instead of a

constant value of wavelength k. In this paper, we refer to refractive index analysis

undertaken using this dispersion equation as ‘the original Swanepoel method’ (Swanepoel

1985). The Cauchy-like model in Eq. (1) assumed that the material band gap, where a

resonance in refractive index occurs, is at zero frequency and the model is thus not correct

if bandgaps or vibrational absorption bands fall within or close to the frequency range of

interest as is the case with chalcogenide glasses (Dantanarayana 2012; Dantanarayana et al.

2014).

In order to extend the Swanepoel method to chalcogenide glasses operating transpar-

ently in the MIR spectral region, another refractive index model is needed to replace the

Cauchy model as the dispersive equation. Dantanarayana et al. (2014) demonstrated that a

two-term Sellmeier model is sufficient to describe the refractive index dispersion of

chalcogenide glasses in the transparent region (where k2 � n2). Following Dantanarayana

et al. (2014), the transparent region of As40Se60 and GeAsSe was taken to be where the

extinction coefficient k was smaller than 0.0005, which for the two glasses studied in that

0.50.55

0.60.65

0.70.75

0.80.85

0.90.95

1

2 7 12 17 22 27

Tran

smitt

ance

Wavelength /microns

normal incidence

30 degree incidence

0.4

0.5

0.6

0.7

0.8

0.9

1

2 7 12 17 22 27

Tran

smitt

ance

Wavelength /microns

normal incidence

30 degree incidence

(a)

(b) The second overtone of the fundamental stretching band

Fig. 2 Transmission spectra for: a a 20 lm target thick pressed Ge16As24Se15.5Te44.5 sample at normalincidence (solid curve, blue online) and at an incident angle of 30� (dotted curve, red online) and b a 25 lmtarget thick pressed As40Se60 sample at normal incidence (solid curve, blue online) and at an incident angleof 30� (dotted curve, red online). (Color figure online)


123

paper was the region extending from *0.7 to *30 lm wavelength. Figure 3 shows the

absorption coefficient of our Ge16As24Se15.5Te44.5 glass, measured using spectroscopic

ellipsometry, from which the transparent region is determined as the wavelength range

*1.5 to *29 lm. Unlike the Cauchy method, the Sellmeier model can be used to model

materials with one bandgap and one vibrational absorption band in the frequency range of

interest. The Sellmeier relation is defined as:

n2ðkÞ ¼ A0 þXN

n¼1

Ank2

k2 � a2n

ð2Þ

where A0 and An are dimensionless coefficients, N is the number of Sellmeier coefficients

(here 2), an indicates the N material resonant absorption wavelengths assumed in the

model, and k is wavelength.

Because of the theoretically unbounded nature of the material resonances in the Sell-

meier model (i.e. the right hand side of Eq. (2) becomes infinite as k ! an), the model can

only be applied to the non-absorbing spectral region of the material, defined here as the

transparent region. In the present paper, the original Swanepoel method was improved by

changing the dispersive model from a Cauchy model to a two-term Sellmeier model in

order to determine better the thickness and refractive index of thin films of chalcogenide

glass in the mid-infrared region.

3.2 The theory of the improved Swanepoel method and analysis

It was assumed that the refractive dispersion is described by a two-term Sellmeier equation

[when N = 2 in Eq. (2)] and that the refractive index was isotropic. It follows that the

spectral dependence of the refractive index could be accurately represented by five con-

stants a1, a2, A0, A1 and A2 in the region where the extinction coefficient k satisfied

k2 � n2.

For normal incidence on a thin film with thickness d, the wavelengths of the trans-

mission extrema are given by:

2n0d ¼ mk0 ð3Þ

where k0 is the wavelength at the maxima or minima of the normal incidence transmission

spectrum, n0 is the refractive index of the material at k0, and m is the order number which

has integer values for the maxima and half-integer values for the minima in the case where

00.20.40.60.8

11.21.41.61.8

2

0 5 10 15 20 25 30 35

Extin

ctio

n co

effe

cien

t k

Wavelength /μm

Fig. 3 Extinction coefficientk variation with wavelengthdetermined from spectroscopicellipsometry measurement on abulk sample of ourGe16As24Se15.5Te44.5 glass


123

the thin film has no substrate (as in the present study) or the refractive index of thin film is

larger than that of the substrate.

Only m and the product n0d can be determined solely from Eqs. (2) and (3); thus

another experimental equation had to be found to determine the other constants. This was

obtained by measuring a second transmission spectrum at an angle of incidence i[ 0, as

shown in Fig. 2. The equation for the interference extrema then became (Crawford 1968):

2nid cos r ¼ mki; ð4Þ

where r is the angle of the refraction in the film, ki is wavelength at the maxima or minima

of the non-normal incidence transmission spectrum, and ni is the refractive index of the

material at ki. Equation (4) can be used for a stand-alone thin film (as in the present study)

or a film on a transparent substrate.

In Fig. 2a, b, the solid curve is the interference transmission spectrum for normal

incidence from 2 to 26 lm. The dotted curve is the spectrum of the same thin film at an

incidence angle of 30�. Oblique incidence shifts the interference transmission spectrum to

shorter wavelength.

Because the accuracy of the calculation depends on displacement between the inter-

ference spectrum for normal incidence and that for oblique incidence, one way to increase

the displacement, according to Eq. (3) is by decreasing the sample thickness. For the FTIR

thin film measurements, the desired wavelength range was from 2 to 25 lm. In order to

have a complete spectrum over this wavelength range, the sample thickness had to be

larger than 5 lm for most of chalcogenide glasses (for which n is typically in the range

from 2.2 to 3.8). However, the thinner the thickness of the thin films the more difficult it

became to collect a complete thin-film sample because of a tendency of the hot-pressed

film to stick to the tungsten carbide plates. In our experiments, 20 and 25 lm thick films

provided sufficient displacements between the two sets of fringes and yet were robust

enough to be readily collected from the hot-press.

Following Swanepoel (1985), the order number m is now considered not to be a dis-

crete-order number but a continuous mathematical variable. The spectrum for normal

incidence is shifted towards shorter wavelengths by increasing each order number by an

amount Dm, where Dm can be different for each m. The shifting in extrema due to oblique

incidence, according to Eq. (4), can be written in terms of normal incidence as:

2nid ¼ m þ Dmð Þki ð5Þ

Following Swanepoel (1985) it is noted for later discussion that Eqs. (4), (5), and Snell’s

law (Eq. 7), yield:

cos r ¼ m

m þ Dm; ð6Þ

using:

ni ¼sin i

sin r; ð7Þ

and that it follows from Eq. (3) that two adjacent extrema of the normal-incidence spec-

trum at wavelengths k01 and k02, obey


123

m ¼ n01k02

2 n02k01 � n01k02ð Þ : ð8Þ

Substitution Eq. (2) into Eq. (3) yields:

m2k20 ¼ A þ Bk2

0

k20 � a2

1

þ Ck20

k20 � a2

2

; ð9Þ

where A ¼ 4ad2, B ¼ 4bd2 and C ¼ 4cd2. The order numbers m at each extremum are

required before the values of A, B, C, a1 and a2 in the two-term Sellmeier fit can be

determined.

In Swanepoel (1985) the values of m were found by introducing an intermediate

quantity M as

M ¼ k02

2 k01 � k02ð Þ ; ð10Þ

where M�m. M = m in the non-dispersive case. In the dispersive case, the dispersion of

M was described by the approximate form:

M ¼ Cs

k0

1 þ Ds

kyo

� �ð11Þ

where Cs, Ds and y are fitting parameters. It will be shown in Sect. 4 that this approach

works well for determining m in the present studies. The dispersion of M can also be

described using a Sellmeier-inspired fitting of the form:

M ¼ E

k0

1 þ F

1 � b1

k0

� �2þ G

1 � b2

k0

� �2

0B@

1CA; ð12Þ

where E, F, G, b1 and b2 are fitting parameters which can then be used to find values of m

(BM) where, at each extremum wavelength ko, m takes an integer value close to trans-

mission maxima and half integer values of m close to transmission minima

Substitution Eq. (2) into Eq. (5), using Eq. (9), and solving for Dm yields:

Dm ¼ m2 k20

k2i

þ B

k2i

1

1 � a1

ki

� �2� 1

1 � a1

k0

� �2

0B@

1CAþ C

k2i

1

1 � a2

ki

� �2� 1

1 � a2

k0

� �2

0B@

1CA

264

375

1=2

�m:

ð13Þ

The value of ni at each wavelength can be calculated using Eqs. (13), (6) and (7).

Since m increases by 1/2 for each successive extremum described by Eq. (5), the value

of d, film thickness, can subsequently be determined. If the order the first extremum, m1, is

considered to be at the long-wavelength end, Eq. (5) can be written for the successive

extreme as described in an earlier paper of Swanepoel (1983), as:

L

2þ Dm

� �¼ 2d

ni

ki

� m1; ð14Þ


123

where m1 is the first order number, L = 0, 1, 2, 3…. Using L2þ Dm

� �as the dependent

variable and ni

kias the independent variable, a straight line can be plotted whose gradient

can be used to determine d.

With d and m known, the final values of refractive index at the wavelength of extrema

can be determined from Eq. (3). The two-term Sellmeier model of Eq. (2) is then used to

fit the final values of refractive index.

4 Results and discussion

The interference transmission spectra of the Ge16As24Se15.5Te44.5 and the As40Se60 thin

films are shown respectively in Fig. 2a, b. The solid curves in Fig. 2 represent the inter-

ference transmission spectrum for normal incidence whereas the dotted spectrum is that for

an incidence angle of 30�. Because the accuracy of the calculation depends on (k0 � ki),the angle of non-normal incidence should be large enough to get sufficient displacement

but not so large that the lateral spreading of the beam provides errors in the case of non-

uniform films. No particular significance should be attached to the actual values of

transmission of the oblique angle of incidence spectrum (dotted curve, red online) because

the effects of polarization can influence its magnitude for large angles of incidence (Fowles

1975). Another method to get sufficient displacement is to decrease the thickness of thin

films. As discussed in Sect. 3.2, decreasing the thickness of the thin film will result in

increasing difficulties in collecting complete samples after the hot-pressing. Thin films

having nominal thicknesses of 20 and 25 lm provide sufficient displacements between two

fringes and yet were strong enough to be collected.

Since the interference fringes below 2.85 lm are influenced by atmospheric water

absorption at wavelengths around 2.7 lm and the displacements (k0 � ki) were relatively

small, which led to errors in determining the thickness of the thin film as well as of the

refractive index, the extrema at wavelengths below 2.85 lm were not selected when

determining the thickness of the thin film. Some wavelengths of the extrema of Ge16-

As24Se15.5Te44.5 and As40Se60 thin films are listed in Tables 1 and 2 as k0 for normal

incidence and as ki for oblique incidence. These values could be determined to an accuracy

of 1 nm from the transmission spectra, sufficient to get a meaningful result (Swanepoel

1985).

The values of M that were calculated from Eq. (10), using the wavelengths at which

adjacent extrema occur at normal incidence, are shown in Table 2. These values were fitted

to Eq. (12) iterating b1 in steps of 0.0001 lm from 0 to 1.5 lm and b2 in steps of 0.01 lm

from 32 to 100 lm for Ge16As24Se15.5Te44.5, since the resonance wavelength b1 and b2

must be initially placed outside the transparent window fitting region to avoid the Sellmeier

model becoming unbounded (Dantanarayana et al. 2014). For AsSe, b1 was varied from 0

to 0.7 lm in steps of 0.0001 lm and b2 was iterated from 30 to 100 lm in steps of

0.01 lm. A good fit was obtained for both Ge16As24Se15.5Te44.5 and As40Se60 yielding

estimates for m at each extrema wavelength ko, termed my following the notation of

Swanepoel (1985), as

myk0 ¼ 123:20 þ 16:19

1 � 1:22k0

� �2þ 1:04 � 10�6

1 � 100:00k0

� �2for Ge16As24Se15:5Te44:5 ð15aÞ


123

myk0 ¼ 142:60 þ 519:80

1 � 0:70k0

� �2þ 5:33 � 10�6

1 � 100:00k0

� �2for As40Se60 ð15bÞ

The values of M from Eq. (10) and my from Eqs. (15a, 15b) are shown in Table 1, together

with the actual value of m deduced.

The original Swanepoel approach to determining M, summarised by Eq. (11), yields

Mk0 ¼ 164:3268

k3:60

þ 139:6986 for Ge16As24Se15:5Te44:5 ð16aÞ

Table 1 Calculations of refractive index of Ge16As24Se15.5Te44.5 (GeAsSeTe) and As40Se60 (AsSe) thinfilms from the improved Swanepoel method

Thin film k0 ki M my m Dm ni n nellipsometry

GeAsSeTe 22.786 22.509 6.1 6.1 6 0.0770 3.150 3.131 3.1255

21.052 20.760 7.0 6.6 6.5 0.0944 2.967 3.134 3.1371

19.635 19.370 7.3 7.1 7 0.0985 3.011 3.148 3.1447

18.381 18.136 7.3 7.6 7.5 0.1036 3.040 3.158 3.1503

17.203 16.996 8.2 8.1 8 0.0994 3.201 3.152 3.1549

16.217 16.013 8.7 8.6 8.5 0.1102 3.136 3.157 3.1583

… …3.042 3.005 46.7 46.8 46 0.5650 3.214 3.205 3.2065

3.010 2.974 48.3 47.4 46.5 0.5669 3.227 3.205 3.2072

2.979 2.943 47.7 47.9 47 0.5864 3.195 3.206 3.2078

2.948 2.912 48.4 48.4 47.5 0.5875 3.209 3.207 3.2085

2.918 2.883 49.0 49.0 48 0.5944 3.207 3.208 3.2092

2.888 2.853 49.5 48.5 0.6008 3.206 3.208 3.2099

d = 21.83 lm n3:1lm from the minimum deviation method ¼ 3:1920

AsSe 23.896 23.5169 7.5 7.3 7 0.1213 2.720 2.710 –

22.411 22.0428 8.0 7.8 7.5 0.1333 2.687 2.723 –

21.087 20.7323 8.6 8.2 8 0.1440 2.671 2.733 –

19.934 19.5787 8.3 8.7 8.5 0.1611 2.604 2.745 –

18.803 18.5132 9.7 9.2 9 0.1466 2.804 2.742 –

17.880 17.5843 9.8 9.7 9.5 0.1655 2.714 2.752 –

… …3.040 2.991 57.8 57.6 57 0.9311 2.800 2.807 –

3.014 2.966 57.3 58.1 57.5 0.9341 2.808 2.807 –

2.988 2.941 58.4 58.6 58 0.9296 2.826 2.807 –

2.962 2.916 59.5 59.1 58.5 0.9392 2.824 2.808 –

2.938 2.891 60.7 59.6 59 0.9554 2.812 2.808 –

2.914 2.867 – 60.1 59.5 0.9717 2.800 2.808 –



123

Mk0 ¼ 930:3893

k3:60

þ 173:9862 for As40Se60 ð16bÞ

As in Swanepoel (1985) the values of my are calculated from the first terms on the right

hand side of Eqs. (16a) and (16b), and are shown in Table 2 along with the connected

values of m. From Tables 1 and 2, it can be seen that both of these methods are able to

determine the actual order number m of the fringes in Fig. 2.

The actual values of m are used in Eq. (9), iterating a1 in step of 0.0001 lm from 0 to

1.5 lm and a2 in step of 0.01 lm from 32 to 100 lm for Ge16As24Se15.5Te44.5. For

As40Se60, b1 was iterated from 0 to 0.7 lm in step of 0.0001 lm and b2 was iterated from

30 to 100 lm in step of 0.01 lm. The best fit was obtained for a1 ¼ 0:4051 and a2 ¼ 41:85

Table 2 Calculations of the refractive index of Ge16As24Se15.5Te44.5 (GeAsSeTe) and As40Se60 (AsSe) thinfilms from the original Swanepoel method

Thin film k0 ki M my m Dm ni n nellipsomtry

GeAsSeTe 22.786 22.509 6.1 6.1 6 0.0740 3.212 3.158 3.1255

21.052 20.760 7.0 6.6 6.5 0.0916 3.010 3.159 3.1371

19.635 19.370 7.3 7.1 7 0.0963 3.045 3.159 3.1447

18.381 18.136 7.3 7.6 7.5 0.1017 3.067 3.160 3.1503

17.203 16.996 8.2 8.1 8 0.0980 3.224 3.160 3.1549

16.217 16.013 8.7 8.6 8.5 0.1089 3.154 3.161 3.1583

…3.042 3.005 46.7 45.9 46 0.5660 3.217 3.206 3.2065

3.010 2.974 48.3 46.4 46.5 0.5673 3.230 3.206 3.2072

2.979 2.943 47.7 46.9 47 0.5853 3.198 3.207 3.2078

2.948 2.912 48.4 47.4 47.5 0.5862 3.212 3.207 3.2085

2.918 2.883 49.0 47.9 48 0.5930 3.210 3.208 3.2092

2.888 2.853 48.4 48.5 0.5993 3.210 3.209 3.2099


AsSe 23.896 23.5169 7.5 7.3 7 0.1131 2.815 2.761 –

22.411 22.0428 8.0 7.8 7.5 0.1258 2.765 2.761 –

21.087 20.7323 8.6 8.3 8 0.1371 2.735 2.761 –

19.934 19.5787 8.3 8.7 8.5 0.1546 2.657 2.761 –

18.803 18.5132 9.7 9.3 9 0.1415 2.853 2.762 –

17.880 17.5843 9.8 9.7 9.5 0.1604 2.755 2.762 –

…3.040 2.991 57.8 57.2 57 0.9394 2.788 2.797 –

3.014 2.966 57.3 57.7 57.5 0.9323 2.796 2.797 –

2.988 2.941 58.4 58.2 58 0.9379 2.814 2.798 –

2.962 2.916 59.5 58.7 58.5 0.9475 2.812 2.798 –

2.938 2.891 60.7 59.2 59 0.9639 2.800 2.798 –

2.914 2.867 – 59.7 59.5 0.9803 2.788 2.799 –



123

in the case of Ge16As24Se15.5Te44.5, which is shown in Fig. 3a and a1 ¼ 0:6584 and a2 ¼58:30 in the case of As40Se60 as shown in Fig. 3c yielding:

m2k20 ¼ 1:118 þ 1:924 � 104k2

k2 � 0:40512þ 1353k2

k2 � 41:852for Ge16As24Se15:5Te44:5 ð17aÞ

m2k20 ¼ 2:425 � 104 þ 5524k2

k2 � 0:65842þ 8832k2

k2 � 58:302for As40Se60 ð17bÞ

From the original Swanepoel method, the best fit was obtained when x = 1 yielding:

m2k20 ¼ 2:028 � 103

k1þ 1:893 � 104 for Ge16As24Se15:5Te44:5 ð18aÞ

m2k20 ¼ 3:506 � 103

k1þ 2:901 � 104 for As40Se60 ð18bÞ

For Ge16As24Se15.5Te44.5, the best fitting from the original Swanepoel method is shown

in Fig. 4c. The value of R2 (correlation coefficient) is 0.8997 and MSE (mean squares

error) is 3.584 9 103. After the improvement of the method, the fitting is shown in Fig. 4a

18400

18600

18800

19000

19200

19400

19600

19800

0 5 10 15 20 25

(mλ 0

)2

Wavelength /μm

27500

28000

28500

29000

29500

30000

30500

0 10 20 30

(mλ 0

)2

Wavelength /μm

18600

18800

19000

19200

19400

19600

19800

0 0.1 0.2 0.3 0.4

(m*λ

)^2

1/λ

27500

28000

28500

29000

29500

30000

30500

0 0.1 0.2 0.3 0.4

(m* λ

)^2

1/λ

(a) (b)

(c) (d)

Fig. 4 a, b Best fit to Eq. (9) in the improved Swanepoel method, Eqs. (17a, 17b), for a Ge16As24Se15.5-

Te44.5, b As40Se60; c, d best fit to the experimental data using the original Swanepoel method, Eqs. (18a,18b), c Ge16As24Se15.5Te44.5, d As40Se60


123

with R2 = 0.9954 and MSE = 1.621 9 102. Applying the original Swanepoel method to

As40Se60, R2 of the best fit is 0.6702 and the MSE is 4.686 9 104, as shown in Fig. 4d.

From Fig. 4b, the best fit from the improved Swanepoel method yields R2 = 0.9980 and

MSE = 2.858 9 102. The poor fits from the original Swanepoel method indicate that the

original dispersive model is not appropriate to describe the refractive index dispersion of

chalcogenide thin films in the mid-infrared region. In contrast, good quality fitting to the

mk0 data, as in Eq. (9), was obtained by using a two-term Sellmeier model as the dis-

persive equation.

Dm was calculated from Eq. (13) and ni was calculated from Eqs. (6) and (7). The

results of Dm and ni for Ge16As24Se15.5Te44.5 and As40Se60 from the improved method are

shown in Table 1. Figure 5 shows a plot of l/2 ? Dm as a function of ni/ki; Eq. (14) shows

that the gradient of this straight line plot is 2d. The slope of the straight line is 43.66 lm for

Ge16As24Se15.5Te44.5 and 61.76 lm for As40Se60, yielding values for d of d ¼ 21:83 lm

(target thickness: 20 lm) and d ¼ 30:88 lm (target thickness: 25 lm), respectively. The

difference between the calculated thickness and the target thickness can be attributed to the

quality of the shims, short hot-pressing time or the high glass-melt viscosity during the hot

pressing.

The results of Dm and ni for Ge16As24Se15.5Te44.5 and As40Se60 from the original

Swanepoel are shown in Table 2. The plots of Eq. (14) are shown in Fig. 5b for

-10

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1 1.2

l/2+Δ

m

ni/λi /μm-1 -10

0

10

20

30

40

50

0 0.5 1 1.5

l/2+Δ

m

ni/λi /μm-1

-10

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1 1.2

l/2+Δ

m

ni/λi /μm-1-10

0

10

20

30

40

50

60

70

0 0.5 1 1.5

l/2+Δ

m

ni/λi /μm-1

(a) (b)

(c) (d)

Fig. 5 Plot of l/2 ? Dm as a function of ni=ki for: a Ge16As24Se15.5Te44.5 thin film from improvedSwanepoel method; b Ge16As24Se15.5Te44.5 thin film from the original method; c As40Se60 thin film fromimproved Swanepoel method and d As40Se60 thin film from the original Swanepoel method to determine thefilm thickness d


123

Ge16As24Se15.5Te44.5 and Fig. 5d for As40Se60. The calculated thickness of GeAsSeTe and

AsSe are 21.84 and 30.98 lm, respectively.

From Fig. 5a, b, the MSE from the improved method for the Ge16As24Se15.5Te44.5

measurement was 4.302 9 10-4 lower than that from the original Swanepoel method. The

MSE from the improved method for AsSe was 2.425 9 10-3 lower than that from the

original method, as shown in Fig. 5c, d. Although the quality of the line fitting was similar,

the values of Dm from the original method were obtained from the poor fitting in Fig. 4c,

which would have led to an error in determining the film thickness d. In the case of

Ge16As24Se15.5Te44.5, the thickness determination was better by 0.01 lm in the improved

method, which would have led to *0.001 difference in refractive index. In the case of

As40Se60, the observed difference of thickness was 0.1 lm which would have led to a

refractive index difference of *0.009.

With d and m known, the final values of refractive index at the wavelength of extremes,

which are shown in Table 1, could be determined from Eq. (3). The fits of these final

values to Eq. (3) from the improved Swanepoel method are shown in Fig. 6a, b, which

yield:

n2 ¼ 5:259 þ 4:837k2

k2 � 0:57712þ 0:7964k2

k2 � 43:722for Ge16As24Se15:5Te44:5

3.12

3.14

3.16

3.18

3.2

3.22

3.24

0 5 10 15 20 25

Refr

activ

ein

dex

Wavelength /μm

3.12

3.14

3.16

3.18

3.2

3.22

3.24

Refr

activ

ein

dex

Wavelength /μm

2.7

2.72

2.74

2.76

2.78

2.8

2.82

2.84

0 5 10 15 20 25

Refr

activ

ein

dex

Wavelength /μm

2.68

2.7

2.72

2.74

2.76

2.78

2.8

2.82

2.84

0 5 10 15 20 25

Refr

activ

e in

dex

Wavelength /μm

(a) (b)

(c) (d)

0 5 10 15 20 25

Fig. 6 The fits to the final refractive index values (temperature of the measurements: 23.1 ± 0.5 �C) of:a Ge16As24Se15.5Te44.5 from the improved Swanepoel method; b Ge16As24Se15.5Te44.5 from the originalSwanepoel method; c As40Se60 from the improved Swanepoel method and d As40Se60 from the originalSwanepoel method


123

n2 ¼ 3:463 þ 4:341k2

k2 � 0:39202þ 2:124k2

k2 � 56:312for As40Se60

In our previous study of a Sellmeier refractive index model fit to As40Se60 ellipsometry

data (Dantanarayana et al. 2014), several different sets of Sellmeier coefficients could be

obtained depending on the choice of an parameters in the Sellmeier fit. We believe that it is

important that the two-term Sellmeier model lends itself to a physical interpretation as this

could open the way to the model being a predictive tool. In our previous work, in the two-

term Sellmeier the a1 coefficient was proposed to be linked to the optical bandgap and a2

proposed to be associated with vibrational band resonant fundamental absorption. The

value of a2 in Dantanarayana et al. (2014) was 41.395 lm which fits well with the position

of the fundamental As–Se stretching band calculated to occur at 41.7 lm (Maklad et al.

1974). Performing a fit to the present experimental data for the refractive index of As40-

Se60, but now fixing the value of a2 to that which was used in Dantanarayana et al. (2014),

yields:

n2 ¼ 4:196 þ 3:602k2

k2 � 0:438342þ 0:975k2

k2 � 41:3952for As40Se60

with R2 = 0.9975. Compared to the best fit in Fig. 6c, the R2 decreases from 0.9987 to

0.9975. The difference between the two fits may be attributed to the limited experimental

wavelength region (2–25 lm) and the effect of the material absorption [at 20.8 lm—

which is the first overtone of the As–Se fundamental stretching vibrational absorption at

41.7 lm (Maklad et al. 1974)]. For the Ge16As24Se15.5Te44.5 glass no bandgap information

is available, and no obvious material absorption was noticed in the transmission spectrum

and thus a best Sellmeier fit to the experimental data is presented.

The fits of these final values to Eq. (3) from the original Swanepoel method are shown

in Fig. 6b, d, which yield:

n2 ¼ 1:064

k1þ 9:926 for Ge16As24Se15:5Te44:5

n2 ¼ 0:7057

k1þ 7:591 for As40Se60

For the Ge16As24Se15.5Te44.5 thin film, the Sellmeier dispersive equation introduced in

this paper also provided better curve fitting to the refractive index data than the original

dispersive equation, as shown in Fig. 6a, b. The value of R2 improved from 0.8983 to

0.9953. For the As40Se60 thin film, the fit from the improved Swanepoel method is far

better than that from the original method, as shown in Fig. 6c, d. In Fig. 6b, d, it is obvious

that the original dispersive equation could not encompass the entire transparent window of

chalcogenide glasses. The results confirm the findings of Dantanarayana et al. (2014) that a

two-term Sellmeier model, with one resonant absorption in the glass optical bandgap

region and one resonant absorption in the fundamental absorption spectral region in the

mid-infrared, is sufficient for fitting the refractive index dispersion across the transparent

window of each glass.

The refractive indices of Ge16As24Se15.5Te44.5 and As40Se60 from the present thin film

measurements, a prism minimum deviation measurement (Fang et al. 2016) and ellip-

sometry are shown in Fig. 7. All the Ge16As24Se15.5Te44.5 measurements were made on

glass samples from the same glass-melt-batch. The prism and thin film measurements for


123

As40Se60 were also made on glass from the same glass-melt-batch but the ellipsometry

measurements are those reported in Dantanarayana et al. (2014) and were made on

As40Se60 glass from a different glass-melt-batch. The minimum deviation method in our

laboratory is able to determine the refractive index of chalcogenide prisms with a standard

deviation of less than 0.001; we take these results as our benchmark value. From Fig. 7a,

the difference in refractive index for the Ge16As24Se15.5Te44.5 from the thin film mea-

surements and from the minimum deviation measurements is 0.0119 (0.37%) at a wave-

length of 3.1 lm, which might be attributed to their different thermal history (that is, the

difference in quenching and annealing of the bulk glass to make the prism and after hot-

pressing to make the thin film). In contrast, since the thin film and ellipsometry sample of

Ge16As24Se15.5Te44.5 were actually both hot-pressed in a similar manner, which provided a

similar thermal history, this provides understanding of why the error in refractive index

between them is less than 0.17% from 2 to 25 lm. For As40Se60, the difference in

refractive index between the thin film measurements and ellipsometry is greater than that

for the Ge16As24Se15.5Te44.5, but is still less than 1.05% from 2 to 25 lm, and is subject to

unintentional compositional differences between the different glass batches which were

prepared and then melted independently of one another. From Fig. 7b, the difference in

refractive index between the thin film measurements and the prism minimum deviation

measurements is 0.0079 (0.28%) at a wavelength of 3.1 lm, and the difference in the

refractive index from the improved Swanepoel method compared to the commercial Schott

data (Schott 2013) is less than 0.0043. Repeated measurements showed that the refractive

index dispersion is determined by the improved method with a standard deviation of less

than 0.0027 for the As40Se60 thin films and less than 0.002 for the Ge16As24Se15.5Te44.5

thin films.

The Swanepoel method in both the original and improved forms is sensitive to the

calculated sample thickness, as indicated by Eq. (3). The surfaces of the thin films of

Ge16As24Se15.5Te44.5 and As40Se60 produced were both not perfect. However, according to

(Swanepoel 1985) these kinds of glass surface undulation and glass surface defects (so long

as they are limited in size relative to the wavelength of the light) do not affect the

wavelength values of the extrema and hence the method used still yields correct and

accurate values for refractive index and thickness.

3.083.1

3.123.143.163.18

3.23.223.243.26

0 5 10 15 20 25 30

Refr

activ

e in

dex

Wavelength /μm

2.662.68

2.72.722.742.762.78

2.82.822.84

0 5 10 15 20 25 30

Refr

activ

e in

dex

Wavelength /μm

Sellmeier fit to thin film data Sellmeier fit to thin film data

Ellipsometry data of AsSe

Schott data (2012) (a) (b)

Ellipsometry data of GeAsSeTe

Prism measurement Prism measurement

Fig. 7 Sellmeier fit to the final refractive index values of a Ge16As24Se15.5Te44.5 and b As40Se60 from FTIRthin film measurements comparing to the data from minimum deviation measurement, ellipsometry andSchott (2013)


123

5 Conclusions

A hot-pressing technique was used to fabricate chalcogenide glass thin films of 20 and

25 lm target thickness from fibres. The characteristic interference fringes in the spectral

transmission of these thin films were obtained over the wavelength range from 2 to 25 lm

by FTIR spectroscopy. The Swanepoel method (Swanepoel 1985) was modified to include

a two-term Sellmeier model as the dispersive equation and was successfully applied to

determine the refractive index of Ge16As24Se15.5Te44.5 and As40Se60 thin films over the

wavelength range from 2 to 25 lm to an accuracy of better than 0.4%. It was shown that

the original Swanepoel method (Swanepoel 1985) failed to correctly determine the

refractive index dispersion of the chalcogenide glasses studied in the MIR spectral region

because it uses an inappropriate Cauchy-like dispersive equation. The average thicknesses

of the non-uniform thin films were also determined by the improved Swanepoel method

presented.

Acknowledgements The authors are grateful to the Naval Research Laboratory (Washington D. C. USA)for their support in providing the ICL used in this work.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Inter-national License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changes were made.

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